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| In [[mathematics]], more specifically in [[abstract algebra]], the '''commutator subgroup''' or '''derived subgroup''' of a [[group (mathematics)|group]] is the [[subgroup]] [[generating set of a group|generated]] by all the [[commutator]]s of the group.<ref>{{harvtxt|Dummit|Foote|2004}}</ref><ref>{{harvtxt|Lang|2002}}</ref>
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| The commutator subgroup is important because it is the smallest [[normal subgroup]] such that the [[quotient group]] of the original group by this subgroup is [[abelian group|abelian]]. In other words, ''G''/''N'' is abelian if and only if ''N'' contains the commutator subgroup. So in some sense it provides a measure of how far the group is from being abelian; the larger the commutator subgroup is, the "less abelian" the group is.
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| == Commutators ==
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| {{main|commutator}}
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| For elements ''g'' and ''h'' of a group ''G'', the [[commutator]] of ''g'' and ''h'' is <math> [g,h] := g^{-1}h^{-1}gh </math>. The commutator [''g'',''h''] is equal to the identity element ''e'' if and only if ''gh'' = ''hg'', that is, if and only if ''g'' and ''h'' commute. In general, ''gh'' = ''hg''[''g'',''h''].
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| An element of ''G'' which is of the form [''g'',''h''] for some ''g'' and ''h'' is called a commutator. The identity element ''e'' = [''e'',''e''] is always a commutator, and it is the only commutator if and only if ''G'' is [[abelian group|abelian]].
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| Here are some simple but useful commutator identities, true for any elements ''s'', ''g'', ''h'' of a group ''G'':
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| * <math>[g,h]^{-1} = [h,g].</math>
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| * <math>[g,h]^s = [g^s,h^s]</math>, where <math>g^s = s^{-1}gs</math>, the [[commutator#Identities|conjugate]] of g by s.
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| * For any homomorphism ''f'': ''G'' → ''H'', ''f''([''g'',''h'']) = [''f''(''g''),''f''(''h'')].
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| The first and second identities imply that the set of commutators in ''G'' is closed under inversion and under conjugation. If in the third identity we take ''H'' = ''G'', we get that the set of commutators is stable under any endomorphism of ''G''. This is in fact a generalization of the second identity, since we can take ''f'' to be the conjugation automorphism <math> x \mapsto x^s </math>.
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| However, the product of two or more commutators need not be a commutator. A generic example is [''a'',''b''][''c'',''d''] in the [[free group]] on ''a'',''b'',''c'',''d''. It is known that the least order of a finite group for which there exists two commutators whose product is not a commutator is 96; in fact there are two nonisomorphic groups of order 96 with this property.<ref>{{harvtxt|Suárez-Alvarez}}</ref>
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| == Definition ==
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| This motivates the definition of the '''commutator subgroup''' [''G'',''G''] (also called the '''derived subgroup''', and denoted ''G′'' or ''G''<sup>(1)</sup>) of ''G'': it is the subgroup [[generating set of a group|generated]] by all the commutators.
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| It follows from the properties of commutators that any element of [''G'',''G''] is of the form
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| :<math>[g_1,h_1] \cdots [g_n,h_n] </math>
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| for some natural number ''n'', where the ''g''<sub>''i''</sub> and ''h''<sub>''i''</sub> are elements of ''G''. Moreover, since <math> ([g_1,h_1] \cdots [g_n,h_n])^s = [g_1^s,h_1^s] \cdots [g_n^s,h_n^s]</math>, the commutator subgroup is normal in ''G''. For any homomorphism ''f'': ''G'' → ''H'',
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| :<math> f([g_1,h_1] \cdots [g_n,h_n]) = [f(g_1),f(h_1)] \cdots [f(g_n),f(h_n)] </math>,
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| so that <math>f([G,G]) \leq [H,H]</math>.
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| This shows that the commutator subgroup can be viewed as a [[functor]] on the category of groups, some implications of which are explored below. Moreover, taking ''G'' = ''H'' it shows that the commutator subgroup is stable under every endomorphism of ''G'': that is, [''G'',''G''] is a [[fully characteristic subgroup]] of ''G'', a property which is considerably stronger than normality.
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| The commutator subgroup can also be defined as the set of elements ''g'' of the group which have an expression as a product ''g'' = ''g''<sub>1</sub> ''g''<sub>2</sub> ... ''g<sub>k</sub>'' that can be rearranged to give the identity. | |
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| === Derived series ===
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| This construction can be iterated:
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| :<math>G^{(0)} := G</math>
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| :<math>G^{(n)} := [G^{(n-1)},G^{(n-1)}] \quad n \in \mathbf{N}</math>
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| The groups <math>G^{(2)}, G^{(3)}, \ldots</math> are called the '''second derived subgroup''', '''third derived subgroup''', and so forth, and the descending [[normal series]]
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| :<math>\cdots \triangleleft G^{(2)} \triangleleft G^{(1)} \triangleleft G^{(0)} = G</math>
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| is called the '''derived series'''. This should not be confused with the '''[[lower central series]]''', whose terms are <math>G_n := [G_{n-1},G]</math>, not <math>G^{(n)} := [G^{(n-1)},G^{(n-1)}]</math>.
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| For a finite group, the derived series terminates in a [[perfect group]], which may or may not be trivial. For an infinite group, the derived series need not terminate at a finite stage, and one can continue it to infinite [[ordinal number]]s via [[transfinite recursion]], thereby obtaining the '''transfinite derived series''', which eventually terminates at the [[perfect core]] of the group.
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| === Abelianization ===
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| Given a group ''G'', a [[quotient group]] ''G''/''N'' is abelian if and only if [''G'',''G''] ≤ ''N''.
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| The quotient ''G''/[''G'',''G''] is an abelian group called the '''abelianization''' of ''G'' or ''G'' '''made abelian'''.<ref>{{harvtxt|Fraleigh|1976|p=108}}</ref> It is usually denoted by ''G''<sup>ab</sup> or ''G''<sub>ab</sub>.
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| There is a useful categorical interpretation of the map <math>\varphi: G \rightarrow G^{\operatorname{ab}}</math>. Namely <math>\varphi</math> is universal for homomorphisms from ''G'' to an abelian group ''H'': for any abelian group ''H'' and homomorphism of groups ''f'': ''G'' → ''H'' there exists a unique homomorphism ''F'': ''G''<sup>ab</sup> → ''H'' such that <math>f = F \circ \varphi</math>. As usual for objects defined by universal mapping properties, this shows the uniqueness of the abelianization ''G''<sup>ab</sup> up to canonical isomorphism, whereas the explicit construction ''G'' → ''G''/[''G'',''G''] shows existence.
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| The abelianization functor is the [[adjoint functors|left adjoint]] of the inclusion functor from the category of abelian groups to the category of groups. | |
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| Another important interpretation of <math>G^{\operatorname{ab}}</math> is as ''H''<sub>1</sub>(''G'','''Z'''), the first [[homology group]] of ''G'' with integral coefficients.
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| === Classes of groups ===
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| A group ''G'' is an '''[[abelian group]]''' if and only if the derived group is trivial: [''G'',''G''] = {''e''}. Equivalently, if and only if the group equals its abelianization. See above for the definition of a group's abelianization.
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| A group ''G'' is a '''[[perfect group]]''' if and only if the derived group equals the group itself: [''G'',''G''] = ''G''. Equivalently, if and only if the abelianization of the group is trivial. This is "opposite" to abelian.
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| A group with <math>G^{(n)}=\{e\}</math> for some ''n'' in '''N''' is called a '''[[solvable group]]'''; this is weaker than abelian, which is the case ''n'' = 1.
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| A group with <math>G^{(n)} \neq \{e\}</math> for any ''n'' in '''N''' is called a '''non solvable group'''. | |
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| A group with <math>G^{(\alpha)}=\{e\}</math> for some [[ordinal number]], possibly infinite, is called a '''[[perfect radical|hypoabelian group]]'''; this is weaker than solvable, which is the case α is finite (a natural number).
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| == Examples ==
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| * The commutator subgroup of the [[alternating group]] ''A''<sub>4</sub> is the [[Klein four group]].
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| * The commutator subgroup of the [[symmetric group]] ''S<sub>n</sub>'' is the [[alternating group]] ''A<sub>n</sub>''.
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| * The commutator subgroup of the [[quaternion group]] ''Q'' = {1, −1, ''i'', −''i'', ''j'', −''j'', ''k'', −''k''} is [''Q'',''Q'']={1, −1}.
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| * The commutator subgroup of the [[fundamental group]] π<sub>1</sub>(''X'') of a path-connected topological space ''X'' is the kernel of the natural homomorphism onto the first singular [[homology group]] ''H''<sub>1</sub>(''X'').
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| === Map from Out ===
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| Since the derived subgroup is [[Characteristic subgroup|characteristic]], any automorphism of ''G'' induces an automorphism of the abelianization. Since the abelianization is abelian, inner automorphisms act trivially, hence this yields a map
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| :<math>\mbox{Out}(G) \to \mbox{Aut}(G^{\mbox{ab}})</math>
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| ==See also==
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| *[[solvable group]]
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| *[[nilpotent group]]
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| == Notes ==
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| <references/>
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| == References ==
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| * {{ citation | last1 = Dummit | first1 = David S. | last2 = Foote | first2 = Richard M. | title = Abstract Algebra | publisher = [[John Wiley & Sons]] | year = 2004 | edition = 3rd | isbn = 0-471-43334-9 }}
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| * {{ citation | first1 = John B. | last1 = Fraleigh | year = 1976 | isbn = 0-201-01984-1 | title = A First Course In Abstract Algebra | edition = 2nd | publisher = [[Addison-Wesley]] | location = Reading }}
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| * {{citation | last = Lang | first = Serge | authorlink = Serge Lang | title = Algebra | publisher = [[Springer Science+Business Media|Springer]] | series = [[Graduate Texts in Mathematics]] | year = 2002 | isbn = 0-387-95385-X}}
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| * {{ cite web | url = http://math.stackexchange.com/questions/7811/derived-subgroups-and-commutators | first = Mariano | last = Suárez-Alvarez | title = Derived Subgroups and Commutators }}
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| [[Category:Group theory]]
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| [[Category:Functional subgroups]]
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| [[Category:Articles containing proofs]]
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If it is easy to holiday home that you visit yourself or set free for holidays, it could well be sat empty over a bitterly cold winter months when individuals are less in the climate for our own view on break away in the united kingdom. This is a good chance to get in it and get your maintenance done before the coldest weather arrives. A house that isn't regularly occupied is through a greater risk for various reasons than one that is inhabited full time, in which means you shouldn't neglect your holiday home. Indeed you should give it some additional care and effort. Check each room one by one and write out any maintenance issues. A number of circumstances garden a transparent out too, ready for your spring. Could also be considered good to be able to give your next home a completely new lick of paint.
Its hedge cutting evening. A light trim of box and yew hedges should be adequate but my beech hedge seems to outlive a hard prune. Conifer hedges need frequent attention take care of the them sleek and stylish.
Put up some new trim and art to update a bath room. Wallpaper trim is not expensive, and the wide range of designs can complement any decor it is possible to imagine. It's also simple to set up. Add some simple, inexpensive artwork, and space can be completely become different.
In 97 we never did a trade unless there was confirming quantities. In 98 the volume was there anyway, similar to in 98. But, since then, just about every roll higher has been in what store sales call "low volume" and frankly Certain pay that awful much attention into it any many more. The entire run up of 03 into 04 was accomplished on considerably less volume next the averages might have had us "need". Wait on the volume and miss the automobile. Nope, not gunna do it.
Cold and damp air can also cause other concerns such as damp and mould. When you visit your holiday home, open the windows and let some fresh air in. Surely for security reasons it's very important make certain you shut and lock the windows when you exit. If your bathroom does have damp, you may find that mould has grown on the grouting amongst the bathroom wall tiles. Giving the grout a good clean with bleach, or renewing it may well get your bath room looking fresh again.
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